#15218: incorrect degree of ring class fields
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Reporter: mstreng | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-5.12
Component: number fields | Keywords:
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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The degree function returns incorrect answers for ring class fields over
QQ(zeta_3) and QQ(i).
See also [https://groups.google.com/forum/#!topic/sage-nt/FncDnGsdSVI The
second part of this thread on sage-nt, the part with title "Computing ring
class fields"].
{{{
E=EllipticCurve("19a");
s =
E.heegner_point(-3,2).ring_class_field().galois_group().complex_conjugation()
H=s.domain(); H.absolute_degree()
6
}}}
The output should be 2, since ZZ[sqrt(-3)] has trivial picard group.
The problem is that there is a bug in the degree function for these ring
class fields. The fields themselves seem to be correct.
In the method degree_over_H of the RingClassField class in
sage/schemes/elliptic_curves/heegner.py, the degree is calculated using
the following formula:
{{{
# Let K_c be the ring class field. We have by class field theory
that
# Gal(K_c / H) = (O_K/c*O_K)^* / (Z/cZ)^*.
}}}
However, one should also divide out by units in `O_K^*` other than {+/-
1}. This explains the factor 3 that the degree function is off for
K=QQ(sqrt(-3)). This is also exactly the difference between equation
(7.27) of Cox's book (Primes of the form ...) and exercise 7.30 of the
same book.
--
Ticket URL: <http://trac.sagemath.org/ticket/15218>
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